Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains

We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded $C^{1,...

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Bibliographic Details
Main Authors	Lindemulder, Nick, Lorist, Emiel, Roodenburg, Floris, Veraar, Mark
Format	Journal Article
Language	English
Published	17.07.2025
Subjects	Mathematics - Analysis of PDEs Mathematics - Functional Analysis
Online Access	Get full text
DOI	10.48550/arxiv.2507.13478

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Summary:	We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded $C^{1,λ}$-domains with $λ\in[0,1]$, revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity for the corresponding heat equation. This extends the well-posedness theory for parabolic equations to domains with minimal smoothness, where classical methods are inapplicable.
DOI:	10.48550/arxiv.2507.13478